A blood bank wants to determine the least expensive way to transport available b

Question

A blood bank wants to determine the least expensive way to transport available blood donations from Pittsburg and Staunton to hospitals in Charleston, Roanoke, Richmond, Norfolk, and Suffolk. Figure shows the possible shipping paths between cities along with the per unit cost of shipping along each possible arc. Additionally, the courier service used by the blood bank charges a flat rate of $125 any time it makes a trip across any of these arcs, regardless of how many units of blood are transported. The van used by the courier service can carry a maximum of 200 units of blood. Assume that Pittsburg has 600 units of blood type O positive (O+) and 800 units of blood type AB available. Assume that Staunton has 500 units of O+ and 600 units of AB available. The following table summarizes the number of units of each blood type needed at the various hospitals:

Start with a complete LP setup. Setup is followed by a Solver solution.

Parts a through h should be answered on the same worksheet Tab.

Parts i, j and k are three variations of the base Model. You must present parts i, j, and k each on a separate Tab as Solver will be used for each modified model.

This problem is similar to one of the problems covered in class which you may find to be a useful guide.

Part a Setup a complete Mathematical LP   

Part b   Setup the Solver and find the optimum solution.

Part c Which supply node has unshipped supply?   

The solution you have obtained in part b will be called the base solution. Further modifications will be applied and changes observed/discussed.

The travel time, in HOURS, between Nodes is shown in parentheses in the following list:

Pittsburg Node

Pittsburg-> Charleston (5)                 

Pittsburg->Roanoke (6)

Staunton Node

Staunton->Charleston (3)                  

Staunton->Roanoke (2)

Charleston Node

Charleston->Richmond (2)

Charleston->Roanoke (4)

Charleston->Norfolk (3)

Richmond Node

Richmond->Norfolk (2)

Roanoke Node

Roanoke->Norfolk (5)

Roanoke->Suffolk (5)

Norfolk Node

Norfolk->Suffolk (0.5)

Part d How fast supply can reach Suffolk? Use the base solution?

For parts e through k, make changes in the Solver setup and Solve. You are not asked to re-write the LP Mathematical model.

Part e If there is a supply disruption between Norfolk & Suffolk and you have to re-route shipments, what is the optimum cost?   

Part f What is the impact of supply disruption in Part e on travel time for shipments coming into Suffolk as compared to shipping time in part d? Explain the reasons for this difference?   

Part g Explain the difference in cost calculated in part b vs cost calculated in part e.

Part h With the travel times between Nodes given as they are, can the supply get to Suffolk any faster, anyhow? If yes, following what route?

Copy the Solver setup from part b onto a new Tab and modify it for part i,

Part i Charleston is both, a demand node & a transshipment node. Demand at Charleston is 300 units. Due to equipment problems, Charleston can ONLY process a maximum of 800 units meaning if they receive 800 units, they will keep 300 and ship out 500. If they receive 801 units, they will keep 300, ship out 500 and will not be able to ship out unit #801. Modify the Solver setup to include this change & solve. What logistical issues arise with this change? Be specific.

Copy the Solver setup from part b onto a new Tab and modify it for part j

Part j To address the issue that arose in part i, another supply route is added which takes supply from Suffolk to Richmond with a unit cost of 6. Supply route connecting Roanoke to Richmond is not available at this time. Modify the Solver setup to include this change & solve. Is all demand met? What is the total optimized shipping cost?   

Copy the Solver setup from part b onto a new Tab and modify it for part k

Part k To address the issue that arose in part i, and subsequent to adding the Suffolk to Richmond Route, the supply route between Roanoke and Richmond is also now available with a unit cost of 5. Modify the Solver setup to include this change & solve. Is all demand met? How does that change the optimum solution in terms of total optimized shipping cost?   

Units Needed Hospital Charleston Roanoke Richmond Norfolk Suffolk 0+ AB 100 100 500 200 150 200 100 300 500 250 Pittsburgh Charleston Richmond $9 Suffolk S4 S6 Staunton S2 Roanoke Norfolk

Explanation / Answer

Let x11 = quantity of O+ received by Charlston in optimal condition . x11 >= 100

x12 = ,, AB ,, ,, ,, ,, x12>= 200

x21 = quantity of O+ received by Roanoke in optimal condition . x21>= 100

x22 = ,, AB ,, ,, ,, x22.=100

x31 = quantity of O+ received by Richmond in optimal condition . x31>= 500

x32 = ,, AB ,, ,, ,, x32.= 300

x41 = quantity of O+ received by Norfolk in optimal condition x41>= 200

x42 = quantity of AB received by Norfolk in optimal condition x42>= 500

x51 = quantity of O+ received by Suffolk in optimal condition x51>= 150

x52 = ,, AB ,, ,, x52>= 250

then ,

x11+x21+x31+x41+x51 <=

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